Timeline for How many random walk steps until the path self-intersects?
Current License: CC BY-SA 4.0
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| Mar 20, 2019 at 18:20 | comment | added | Gerhard Paseman | The version of Joseph's question on a compact space is likely in the graph theory literature. Consider a cover of the space by uniform mostly disjoint regions, and make a graph with a region a vertex and a line joining two regions if a step takes one from one region to another. One can now approximate a random walk by a transition matrix, or use the theory of random walks in graphs to compute a desired expectation. Thus the actual expectation can be considered as a limit of expectations using various partitions of the space. Gerhard "Still Interested In Torus Version" Paseman, 2019.03.20. | |
| Mar 20, 2019 at 17:21 | comment | added | Gerhard Paseman | I would like to point out that many of my signatures encode some double meaning. In this particular case, I want to make it clear that this analysis fails on other manifolds. In particular, Joseph's problem makes sense on a torus or pseudo sphere, even on non-oriented manifolds or graphs with a rigid geometric structure. I encourage people to consider the version for a torus. Gerhard "Promises To Not Grade Work" Paseman, 2019.03.20. | |
| Mar 20, 2019 at 17:15 | history | edited | Gerhard Paseman | CC BY-SA 4.0 |
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| Mar 20, 2019 at 17:08 | history | edited | Gerhard Paseman | CC BY-SA 4.0 |
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| Mar 20, 2019 at 17:03 | history | answered | Gerhard Paseman | CC BY-SA 4.0 |